Method, apparatus, and program for restoring phase information

ABSTRACT

A method of restoring phase information using a phase-contrast method, which can improve estimated accuracy of a phase. This method is a method of restoring phase information by detecting intensity of radiation, and includes the steps of: (a) obtaining three first differential signals representing differentials between one image signal and another image signal based on four image signals obtained by detecting intensity of radiation on four planes and representing radiation image information respectively; (b) obtaining second and third differential signals representing differentials between image signals relative to two directions orthogonal to each other within the planes with respect to the three image signals; (c) obtaining a Laplacian of phase based on the three image signals and three sets of the first to third differential signals; and (d) performing inverse Laplacian operation on the Laplacian of phase so as to restore phase information.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a method, an apparatus, and aprogram for restoring phase information used for constructing an imagebased on image information obtained by radiation imaging etc. Note that,in this application, the term “radiation” indicates radiation in a broadsense including a corpuscular beam such as an electron beam, and anelectromagnetic wave, in addition to common radiation such as an X-ray,an α-ray, a β-ray, a γ-ray, and ultraviolet rays.

[0003] 2. Description of a Related Art

[0004] Conventionally, an imaging method using an X-ray etc. is utilizedin various fields, and particularly, in the medical field, the method isone of the most important means for diagnosis. Since the first X-rayphotograph was implemented, the X-ray photography method has beenrepeatedly improved, and a method using a combination of a fluorescentscreen and an X-ray film is in the mainstream at present. On the otherhand, in recent years, various digitized devices applying X-ray CT,ultrasonic waves, MRI, etc. are in practical use, and construction of adiagnostic information processing system etc. in hospitals is beingpromoted. Many studies are also made for digitizing the imaging systemwith regard to X-ray images. Digitizing the imaging system enables tostore a large amount of data for a long period without degradation inimage, and helps to make progress toward the medical diagnosticinformation processing system.

[0005] Now, a radiation image obtained as described above is generatedby converting intensity of the radiation transmitted through an objectinto brightness of the image. For example, when performing imaging on aregion including a bone part, the radiation transmitted through the bonepart is largely attenuated, and the radiation transmitted through aregion other than the bone part, i.e., a soft part is slightlyattenuated. In this case, since the difference in intensity between theradiation transmitted through different tissues is large, ahigh-contrast radiation image can be obtained.

[0006] On the other hand, for example, when imaging a region of the softpart such as a breast, since radiation is easily transmitted through thesoft part as a whole, the difference between tissues in the soft part isdifficult to appear as the difference in intensity of the transmittedradiation. Therefore, only a low-contrast image can be obtained withrespect to the soft part. Thus, the radiation imaging method is notsuitable as a method of visualizing the slight difference betweentissues in the soft part.

[0007] Here, as information included in the radiation transmittedthrough the object, there is phase information in addition to intensityinformation. Recently, a phase-contrast method of generating an imageusing this phase information is under study. The phase-contrast methodis a phase information restoration technology to convert phasedifference produced by an X-ray etc. transmitted through the object intobrightness of an image.

[0008] The phase-contrast method includes techniques for obtaining phasedifference based on interference light produced by using aninterferometer or a zone plate, and for obtaining phase difference basedon diffracted light. Of these techniques, the diffraction technique forobtaining phase difference based on diffracted light is to obtain phasedifference according to the following principle. An X-ray, for example,propagates within a material since a wave progresses similarly to light.The propagation velocity thereof varies in accordance with therefractive index the material has. Therefore, when irradiating an objectwith an X-ray that has a uniform phase, the way the X-ray propagatesvaries in accordance with the difference between tissues in the object.Thereby, the wavefront of the X-ray transmitted through the object isdistorted and, as a result, diffraction fringes are produced on theX-ray image obtained based on the transmitted X-ray. The pattern of thediffraction fringes differs in accordance with the distance between thescreen on which the X-ray image is formed and the object, and thewavelength of the X-ray. Thus, analyzing two or more X-ray images havingdifferent patterns of diffraction fringes, phase differences of theX-ray produced on the respective positions on the screen can beobtained. Converting the phase differences into brightness, an X-rayimage that clearly shows the difference between tissues in the objectcan be obtained.

[0009] Particularly, in the radiation after transmitted through a softpart of the object, since the difference in phase is larger than thedifference in intensity in accordance with the difference betweentissues through which the radiation has been transmitted, the subtledifference between tissues can be visualized by using the phase-contrastmethod.

[0010] In order to use the above phase-contrast method, imagingconditions in radiation imaging or techniques for restoring phase frompatterns of diffraction fringes are under study.

[0011] For example, B. E. Allman et al. “Noninterferometric quantitativephase imaging with soft x rays”, J. Optical Society of America A, Vol.17, No. 10 (October 2000), pp. 1732-1743, discloses that an X-ray imageis constructed by performing phase restoration based on imageinformation obtained by imaging with soft X-rays.

[0012] In this reference, the basic expression of phase restoration, TIE(transport of intensity equation) is used. $\begin{matrix}{{\kappa \frac{\partial{I\left( \overset{\rightharpoonup}{r} \right)}}{\partial z}} = {{- \nabla}\bot{\cdot \left\{ {{{I\left( \overset{\rightharpoonup}{r} \right)}\nabla}\bot{\varphi \left( \overset{\rightharpoonup}{r} \right)}} \right\}}}} & (1)\end{matrix}$

[0013] Where${\nabla\bot} = \left( {\frac{\partial}{\partial x},\quad \frac{\partial}{\partial y}} \right)$

[0014] In addition, κ denotes wave number.

[0015] Here, a principle of the phase restoration will be described byreferring to FIG. 11. As shown in FIG. 11, an X-ray having a wavelengthλ is output from the left side of the drawing, transmitted through anobject plane 101 and enters a screen 102 at a distance of z from theobject plane 101. Assuming that the intensity and the phase of the X-rayat a position (x,y) on the screen 102 are I(x,y) and φ(x,y),respectively, a relationship between intensity I(x,y) and phase φ(x,y)is expressed by the following expression. Here, the intensity I issquare of amplitude of wave. $\begin{matrix}{{\frac{2\pi}{\lambda}\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z}} = {{- \nabla} \cdot \left\{ {{I\left( {x,\quad y} \right)}{\nabla{\varphi \left( {x,\quad y} \right)}}} \right\}}} & (2)\end{matrix}$

[0016] Substituting κ=2π/λ into expression (2) and rewriting (x,y)components into vector r, TIE expressed by expression (1) is derived.

[0017] However, since the above TIE is difficult to be solved, TIE hasbeen used mainly by performing approximation thereon. For example, T. E.Gureyev et al. “Hard X-ray quantitative non-interferometricphase-contrast imaging”, SPIE Vol. 3659 (1999), pp. 356-364, disclosesthat an X-ray image is constructed by performing phase restoration basedon image information obtained by imaging with hard X-rays.

[0018] In this reference, TIE expressed by expression (1) isapproximated as follows. First, expression (1) is developed. Note that,in the following expressions, the vector r in the above reference isrewritten into (x,y) components. $\begin{matrix}{{{- \kappa}\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z}} = {\left( {\frac{\partial\quad}{\partial x},\quad \frac{\partial}{\partial y}} \right) \cdot \left( {{{I\left( {x,\quad y} \right)}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial x},\quad {I\left( {x,\quad y} \right)}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial y}} = {{{\frac{\partial}{\partial x}\left( {{I\left( {x,\quad y} \right)}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial x}} \right)} + {\frac{\partial}{\partial y}\left( {{I\left( {x,\quad y} \right)}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial y}} \right)}} = {{{{I\left( {x,\quad y} \right)}\left( {\frac{\partial^{2}{\varphi \left( {x,\quad y} \right)}}{\partial x^{2}} + \frac{\partial^{2}{\varphi \left( {x,\quad y} \right)}}{\partial y^{2}}} \right)} + {\frac{\partial{I\left( {x,\quad y} \right)}}{\partial x}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial x}} + {\frac{\partial{I\left( {x,\quad y} \right)}}{\partial y}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial y}}} = {{{I\left( {x,\quad y} \right)}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}}} + {{\nabla{I\left( {x,\quad y} \right)}} \cdot {\nabla{\varphi \left( {x,\quad y} \right)}}}}}}} \right.}} & (3)\end{matrix}$

[0019] Where$\nabla^{2}{= {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}}}$

[0020] Approximating the second term on the right side of expression (3)to zero, the approximation expression expressed by the followingexpression (4) is obtained. $\begin{matrix}{\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z} \cong {{- \frac{{I\left( {x,\quad y} \right)}\quad}{\kappa}}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}}}} & (4)\end{matrix}$

[0021] In expression (4), φ(x,y) can be obtained from I(x,y) by asolution method such as the finite element method.

[0022] However, in the approximation expression (4), there is a problemthat the estimated accuracy of the phase φ(x,y) becomes low when theapproximation on the second term ∇I(x,y)·∇φ(x,y) included in expression(3) to zero is not appropriate.

[0023] Further, using a differential coefficient of intensity obtainedfrom two pieces of detection data instead of the left side of expression(4), a phase can be restored by using at least two images. However,there is a problem that, when the phase is restored from a small numberof images as many as two, the estimated accuracy of the phase becomeslow in the case where the images are degraded due to noise etc.

SUMMARY OF THE INVENTION

[0024] The present invention has been achieved in view of theabove-described problems. An object of the present invention is toprovide a phase information restoring method that enables to improveestimated accuracy of a phase when constructing a radiation image by thephase-contrast method. Another object of the present invention is toprovide a phase information restoring apparatus and a phase informationrestoring program for implementing the above phase information restoringmethod.

[0025] In order to solve the above-described problems, a phaseinformation restoring method according to a first aspect of the presentinvention is a method of restoring phase information of radiationtransmitted through an object on the basis of an image signal obtainedby detecting intensity of the radiation transmitted through the object,the method comprises the steps of: (a) obtaining at least three firstdifferential signals representing differentials between one image signaland another image signal on the basis of at least four image signalsobtained by detecting intensity of radiation on at least four planesthat are parallel and positioned at different distances from the object,the at least four image signals respectively representing radiationimage information on the at least four planes; (b) obtaining seconddifferential signals and third differential signals representingdifferentials between image signals relative to two directionsorthogonal to each other within the planes with respect to at leastthree image signals; (c) obtaining a Laplacian of phase on the basis ofthe at least three image signals and at least three sets of the first tothird differential signals; and (d) performing inverse Laplacianoperation on the Laplacian of phase so as to restore phase information.

[0026] A phase information restoring apparatus according to the firstaspect of the present invention is an apparatus for restoring phaseinformation of radiation transmitted through an object on the basis ofan image signal obtained by detecting intensity of the radiationtransmitted through the object, the apparatus comprises: differentialprocessing means for obtaining at least three first differential signalsrepresenting differentials between one image signal and another imagesignal on the basis of at least four image signals obtained by detectingintensity of radiation on at least four planes that are parallel andpositioned at different distances from the object, the at least fourimage signals respectively representing radiation image information onthe at least four planes, and second differential signals and thirddifferential signals representing differentials between image signalsrelative to two directions orthogonal to each other within the planeswith respect to at least three image signals; Laplacian processing meansfor obtaining a Laplacian of phase on the basis of the at least threeimage signals and at least three sets of the first to third differentialsignals; inverse Laplacian processing means for performing inverseLaplacian operation on the Laplacian of phase so as to restore phaseinformation.

[0027] A phase information restoring program according to the firstaspect of the present invention is a program used for restoring phaseinformation of radiation transmitted through an object on the basis ofan image signal obtained by detecting intensity of the radiationtransmitted through the object, the program allows a CPU to execute theprocedures of: obtaining at least three first differential signalsrepresenting differentials between one image signal and another imagesignal on the basis of at least four image signals obtained by detectingintensity of radiation on at least four planes that are parallel andpositioned at different distances from the object, the at least fourimage signals respectively representing radiation image information onthe at least four planes; obtaining second differential signals andthird differential signals representing differentials between imagesignals relative to two directions orthogonal to each other within theplanes with respect to at least three image signals; obtaining aLaplacian of phase on the basis of the at least three image signals andat least three sets of the first to third differential signals; andperforming inverse Laplacian operation on the Laplacian of phase so asto restore phase information.

[0028] According to the first aspect of the present invention, sinceapproximation is not performed except on a differential coefficient partof intensity when using the expression of phase restoration,high-accuracy phase restoration can be achieved.

[0029] A phase information restoring method according to a second aspectof the present invention is a method of restoring phase information ofradiation transmitted through an object on the basis of an image signalobtained by detecting intensity of the radiation transmitted through theobject, the method comprises the steps of: (a) obtaining pluraldifferential signals representing differentials between one image signaland another image signal on the basis of at least three image signalsobtained by detecting intensity of radiation on at least three planesthat are positioned at different distances from the object, the at leastthree image signals respectively representing radiation imageinformation on the at least three planes; (b) respectively obtainingLaplacian of phases on the basis of the plural differential signals andone of the at least three image signals; (c) performing inverseLaplacian operation on the Laplacian of phases so as to obtain pluralphases respectively; (d) calculating an average value of the pluralphases obtained at step (c).

[0030] A phase information restoring apparatus according to the secondaspect of the present invention is an apparatus for restoring phaseinformation of radiation transmitted through an object on the basis ofan image signal obtained by detecting intensity of the radiationtransmitted through the object, the apparatus comprises: differentialprocessing means for obtaining plural differential signals representingdifferentials between one image signal and another image signal on thebasis of at least three image signals obtained by detecting intensity ofradiation on at least three planes that are positioned at differentdistances from the object, the at least three image signals respectivelyrepresenting radiation image information on the at least three planes;Laplacian processing means for respectively obtaining Laplacian ofphases on the basis of the plural differential signals and one of the atleast three image signals; inverse Laplacian processing means forperforming inverse Laplacian operation on the Laplacian of phases so asto obtain plural phases respectively; average processing means forcalculating an average value of the plural phases obtained in theinverse Laplacian processing means.

[0031] A phase information restoring program according to the secondaspect of the present invention is a program used for restoring phaseinformation of radiation transmitted through an object on the basis ofan image signal obtained by detecting intensity of the radiationtransmitted through the object, the program allows a CPU to execute theprocedures of: (a) obtaining plural differential signals representingdifferentials between one image signal and another image signal on thebasis of at least three image signals obtained by detecting intensity ofradiation on at least three planes that are positioned at differentdistances from the object, the at least three image signals respectivelyrepresenting radiation image information on the at least three planes;(b) respectively obtaining Laplacian of phases on the basis of theplural differential signals and one of the at least three image signals;(c) performing inverse Laplacian operation on the Laplacian of phases soas to obtain plural phases respectively; (d) calculating an averagevalue of the plural phases obtained in procedure (c).

[0032] According to the second aspect of the present invention,high-accuracy phase restoration can be achieved by restoring pluralphases and averaging the restored plural phases.

BRIEF DESCRIPTION OF THE DRAWINGS

[0033]FIG. 1 is a block diagram showing an X-ray imaging systemincluding a phase information restoring apparatus according to a firstembodiment of the present invention;

[0034]FIG. 2 is a diagram showing a structure of an imaging unit shownin FIG. 1;

[0035]FIG. 3 is a flowchart showing a phase information restoring methodaccording to the first embodiment of the present invention;

[0036]FIG. 4 is an explanatory diagram of a phase information restoringmethod according to a second embodiment of the present invention;

[0037]FIG. 5 is a block diagram showing an X-ray imaging systemincluding a phase information restoring apparatus according to a thirdembodiment of the present invention;

[0038]FIG. 6 is a flowchart showing a phase information restoring methodaccording to the third embodiment of the present invention;

[0039]FIG. 7 is a block diagram showing an X-ray imaging systemincluding a phase information restoring apparatus according to a fourthembodiment of the present invention;

[0040]FIG. 8 is a flowchart showing a phase information restoring methodaccording to the fourth embodiment of the present invention.

[0041]FIG. 9 is a block diagram showing a modified example of the X-rayimaging system shown in FIG. 1;

[0042]FIG. 10 is a diagram showing a structure of a reading unit shownin FIG. 9; and

[0043]FIG. 11 is an explanatory diagram showing a principle of phaserestoration.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0044] Now, referring to the drawings, embodiments of the presentinvention will be described in detail. The same component elements willbe given with the same reference numerals and the descriptions thereofwill be omitted.

[0045]FIG. 1 is a block diagram showing an X-ray imaging systemincluding a phase information restoring apparatus according to a firstembodiment of the present invention. This X-ray imaging system has animaging unit 1 for irradiating the object with an X-ray so as to outputdetection data that represents image information of an object, a phaseinformation restoring apparatus 2 for generating image data based on thedetection data, a display unit 3 for displaying a visible image based onthe image data, and an output unit 4 for printing out the visible imageon a film etc.

[0046]FIG. 2 is a diagram showing a structure of the imaging unit 1. Asa light source 11, it is desirable to use a light source generating aradiation beam that is highly coherent and monochromatic. Here, thehighly monochromatic beam indicates a beam that mainly has a singlewavelength. For this purpose, in the embodiment, a synchrotron radiationsource that generates X-rays is used as the light source 11. Thesynchrotron radiation is light (an electromagnetic wave) that isgenerated by accelerating an electron or bending a traveling directionof an electron. The X-ray generated from the light source 11 istransmitted through an object 10 and enters a sensor 12.

[0047] The sensor 12 detects the incident X-ray. As the sensor 12, atwo-dimensional sensor such as a CCD (charge coupled device) having aplurality of detecting elements that convert intensity of the appliedX-ray into electric signals and output the signals is used. Thedetection signal output from the sensor 12 is amplified by an amplifier15, converted into a digital signal (detection data) by an A/D converter16, and output to the phase information restoring apparatus 2.

[0048] The sensor 12 is held by a holding portion 13. The holdingportion 13 is movably supported on a rail 14. The position of theholding portion 13 is controlled by a control unit, which will bedescribed later, of the phase information restoring apparatus 2, and adistance between the object 10 and the sensor 12 is changed under thecontrol of the control unit. Note that the distance between the object10 and the sensor 12 is referred to as an imaging distance hereinafter.

[0049] Referring to FIG. 1 again, the phase information restoringapparatus 2 has a storage unit 21 for temporarily storing the detectiondata output from the imaging unit 1, a differential processing unit 22for obtaining a differential coefficient between detection data atdifferent imaging distances and a differential coefficient betweendetection data at the same imaging distance, a Laplacian processing unit23 for calculating a value that corresponds to a Laplacian of phase, aninverse Laplacian processing unit 24 for performing inverse Laplacianoperation for phase restoration, an image processing unit 25 forgenerating image data on the basis of the restored phase information,and a control unit 26 for controlling the respective units 21-25 and theimaging distance in the imaging unit 1. The phase information restoringapparatus 2 may be configured with a digital circuit or software and aCPU. With a CPU, the control unit 26 including the CPU processes thedetection data on the basis of a phase information restoring programrecorded on a recording medium 27. As the recording medium 27, aflexible disk, a hard disk, an MO, an MT, a RAM, a CD-ROM, a DVD-ROM,etc. are applicable.

[0050] The display unit 3 is a display device such as a CRT, anddisplays a visible image based on image data that represents the phaseinformation restored by the phase information restoring apparatus 2. Theoutput unit 4 is a laser printer, for example, and prints out thevisible image on a film etc. on the basis of the image data.

[0051] Next, a principle of a phase information restoring methodaccording to the present invention will be described. The phaseinformation restoring method according to the present invention is amethod of constructing a visible image by the phase-contrast method, andthe phase restoration is performed on the basis of plural diffractionfringe images obtained with respect to an object by using the basicexpression of phase restoration, TIE (transport of intensity equation).

[0052] TIE expressed by the following expression (5) is transformed soas to obtain expression (6). $\begin{matrix}{{{{- \kappa}\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z}} = {\nabla{\cdot \left\{ {{I\left( {x,\quad y} \right)}{\nabla{\varphi \left( {x,\quad y} \right)}}} \right\}}}}} & (5) \\{{{- \kappa}\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z}} = {{{{I\left( {x,\quad y} \right)}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}}} + {{\nabla{I\left( {x,\quad y} \right)}} \cdot {\nabla{\varphi \left( {x,\quad y} \right)}}}} = {{{I\left( {x,\quad y} \right)}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}}} + {\frac{\partial{I\left( {x,\quad y} \right)}}{\partial x}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial x}} + {\frac{\partial{I\left( {x,\quad y} \right)}}{\partial y}\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial y}}}}} & (6)\end{matrix}$

[0053] Where I(x,y) is detection data representing intensity ofdiffracted light at a position (x,y) on a plane at a distance of z fromthe object.

[0054] In expression (6), the Laplacian ∇²φ(x,y) and the gradients(∂φ(x,y)/∂x,∂φ(x,y)/∂y) of the phase φ(x,y) to be obtained are unknown.If at least three gradients ∇I=(∂I/∂x,∂I/∂y,∂I/∂z) of the intensity ofthe diffracted light can be obtained, expression (6) can be solved.

[0055] Substituting elements of the gradients ∇I₁ to ∇I₃ of theintensity of the diffracted light into expression (6), it is expressedwith matrices by expression (7). $\begin{matrix}{{- {\kappa \begin{pmatrix}\frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial z}\end{pmatrix}}} = {\begin{pmatrix}{I_{1}\left( {x,\quad y} \right)} & \frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial y} \\{I_{2}\left( {x,\quad y} \right)} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial y} \\{I_{3}\left( {x,\quad y} \right)} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial y}\end{pmatrix}\begin{pmatrix}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}} \\\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial x} \\\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial y}\end{pmatrix}}} & (7)\end{matrix}$

[0056] Expression (7) can be solved using an inverse matrix, forexample.

[0057] As described above, in the embodiment, approximation in TIE isminimized to raise the accuracy of the phase restoration and theoperation is simplified by using a matrix form.

[0058] Next, referring to FIGS. 1-3, the phase information restoringmethod according to the first embodiment of the present invention willbe described. FIG. 3 is a flowchart showing the phase informationrestoring method according to the first embodiment of the presentinvention. In the embodiment, a visible image is constructed by usingdetection data representing six diffraction fringe images taken whilechanging the imaging distance as shown in FIG. 2.

[0059] First, at step S1, X-ray imaging is performed. The sensor 12 ispositioned at the position where the imaging distance is z₁ as shown inFIG. 2 and irradiating the object 10 with an X-ray so as to perform theX-ray imaging. Then, the sensor 12 moved to the position where theimaging distance is (z₁+Δz₁) and the X-ray imaging is performed.Similarly, the X-ray imaging is repeated with the sensor positioned atthe imaging distances of z₂, (z₂+Δz₂), z₃, and (z₃+Δz₃). Thereby, thedetection data representing diffraction fringe images are obtained.

[0060] By the X-ray imaging at step S1, detection data I₁(x,y),I₁′(x,y), I₂(x,y), I₂′(x,y), I₃(x,y), and I₃′(x,y) are sequentiallyinput to the phase information restoring apparatus 2. Here, thedetection data I₁(x,y) represents intensity of the diffracted light atthe position (x,y) on a plane at the imaging distance of z₁. Similarly,the detection data I₁′(x,y), I₂(x,y), I₂′(x,y)), I₃(x,y), and I₃′(x,y)represent intensity of the diffracted light at the positions (x,y) onplanes at the imaging distances of (z₁+Δz₁), z₂, (z₂+Δz₂), z₃, and(z₃+Δz₃), respectively. The detection data are sequentially stored inthe storage unit 21 of the phase information restoring apparatus 2.

[0061] Next, at steps S2-S6, the phase information restoring apparatus 2restores a phase on the basis of the detection data stored in thestorage unit 21.

[0062] First, at step S2, the differential processing unit 22 obtains adifferential coefficient between detection data I_(N) and detection dataI_(N)′ using the following expression (8), where Δz_(N)=z_(N)′−z_(N) andN=1, 2, and 3. $\begin{matrix}{\frac{\partial{I_{N}\left( {x,\quad y} \right)}}{\partial z} = \frac{{I_{N}^{\prime}\left( {x,\quad y} \right)} - {I_{N}\left( {x,\quad y} \right)}}{\Delta \quad z_{N}}} & (8)\end{matrix}$

[0063] Then, at step S3, the Laplacian processing unit 23 obtains thegradients ∂I (x,y)/∂x and ∂I(x,y)/∂y of the detection data at respectivepositions (x,y) on xy plane, and generates matrix A(x,y) with three rowsand three columns as expressed by expression (9). $\begin{matrix}{{A\left( {x,\quad y} \right)} = \begin{pmatrix}{I_{1}\left( {x,\quad y} \right)} & \frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial y} \\{I_{2}\left( {x,\quad y} \right)} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial y} \\{I_{3}\left( {x,\quad y} \right)} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial y}\end{pmatrix}} & (9)\end{matrix}$

[0064] Further, the Laplacian processing unit 23 generates vector D(x,y)expressed by expression (10) on the basis of the differentialcoefficient obtained by expression (8). $\begin{matrix}{\overset{}{D\left( {x,y} \right)} = \begin{pmatrix}\frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial z}\end{pmatrix}} & (10)\end{matrix}$

[0065] Next, at step S4, the Laplacian processing unit 23 derives therelational expression of matrix (11) using the matrix A(x,y) and thevector D (x,y) obtained by expressions (9) and (10). $\begin{matrix}{{- {\kappa \begin{pmatrix}\frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial z}\end{pmatrix}}} = {\begin{pmatrix}{I_{1}\left( {x,\quad y} \right)} & \frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial y} \\{I_{2}\left( {x,\quad y} \right)} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial y} \\{I_{3}\left( {x,\quad y} \right)} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial y}\end{pmatrix}\begin{pmatrix}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}} \\\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial x} \\\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial y}\end{pmatrix}}} & (11)\end{matrix}$

[0066] Further, at step S5, the Laplacian processing unit 23 multipliesboth sides of expression (11) by an inverse matrix of the matrix A(x,y)from the left side as expressed by expression (12) so as to obtainvector Φ(x,y). $\begin{matrix}{\overset{}{\Phi \left( {x,\quad y} \right)} = {{{- {\kappa \begin{pmatrix}{I_{1}\left( {x,\quad y} \right)} & \frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial x} & \frac{{\partial{I_{1}\left( {x,\quad y} \right)}}\quad}{\partial y} \\{I_{2}\left( {x,\quad y} \right)} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial y} \\{I_{3}\left( {x,\quad y} \right)} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial x} & \frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial y}\end{pmatrix}}^{- 1}}\begin{pmatrix}\frac{\partial{I_{1}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{2}\left( {x,\quad y} \right)}}{\partial z} \\\frac{\partial{I_{3}\left( {x,\quad y} \right)}}{\partial z}\end{pmatrix}} = \begin{pmatrix}{\nabla^{2}{\varphi \left( {x,\quad z} \right)}} \\\frac{\partial{\varphi \left( {x,y} \right)}}{\partial x} \\\frac{\partial{\varphi \left( {x,\quad y} \right)}}{\partial y}\end{pmatrix}}} & (12)\end{matrix}$

[0067] The first element of the vector Φ(x,y) corresponds to theLaplacian ∇²φ(x,y) of the phase.

[0068] Then, at step S6, the inverse Laplacian processing unit 24performs inverse Laplacian operation on the Laplacian f(x,y)=∇²φ(x,y)obtained at step S5 so as to obtain phase φ(x,y).

[0069] Here, the inverse Laplacian operation will be described indetail. A Fourier transform of f(x,y) is expressed by the followingexpression (13).

F[f(x,y)]=F[∇²φ(x,y)]=−4π²(u ² +v ²)F[φ(x,y)]  (13)

[0070] Where u and v are spatial frequencies that correspond to x and y.

[0071] Hereby, the phase φ(x,y) is expressed as expression (14).$\begin{matrix}{{\varphi \left( {x,\quad y} \right)} = {F^{- 1}\left\lbrack {{- \frac{1}{4{\pi^{2}\left( {u^{2} + v^{2}} \right)}}}{F\left\lbrack {f\left( {x,\quad y} \right)} \right\rbrack}} \right\rbrack}} & (14)\end{matrix}$

[0072] Using expression (14), the inverse Laplacian operation can beperformed. That is, the restored phase φ(x,y) can be obtained byperforming the Fourier transform of f(x,y), multiplying by{−4π²(u²+v²)}⁻¹ and then performing an inverse Fourier transformthereon.

[0073] Here, a value of {−4²(u²+v²)}⁻¹ may be calculated in advancewithin the range where |u| and |v| are not more than a predeterminedvalue, and used when the operation expressed by expression (14) isperformed. That is, in the case where the predetermined value “const” isset, for |u|, |v|≦const, the value of the following expression is usedin expression (14).

[0074] {−4π²(u²+v²)}⁻¹=(the value calculated in advance) For |u|,|v|>const, the value of the following expression is used in expression(14).

−4π²(u ² +v ²)}⁻¹=0

[0075] Thereby, the inverse Laplacian operation can be performed at highspeed.

[0076] Next, at step S7, the image processing unit 25 generates imagedata on the basis of the restored phase φ(x,y). That is, the imageprocessing unit 25 converts the phase φ(x,y) in each pixel into datarepresenting brightness, and performs necessary image processing such asgradation processing and interpolation processing, etc.

[0077] At step S8, the display unit 3 and the output unit 4 display avisible image on a screen, a film, etc. on the basis of the image datagenerated as described above.

[0078] Although, in the embodiment, the method of restoring phase byusing three differential coefficients obtained from six interferencefringe images taken while changing the imaging distance is described,the phase restoration may be performed by using four or moredifferential coefficients obtained from seven or more interferencefringe images. Alternatively, with respect to expression (11), the phaserestoration may be performed on the basis of the vector Φ(x,y) that isobtained by using the least-squares method as expressed by expression(15).

{right arrow over (Φ)}=−κ(A ^(t) A)⁻¹ A ^(t) {right arrow over(D)}  (15)

[0079] Further, as expressed by the following expression (16), only therequired part for obtaining ∇²φ(x,y) among the components in expression(11) maybe calculated without using the inverse matrix. $\begin{matrix}{{\nabla^{2}\varphi} = {{- \kappa}\frac{{K_{1}\frac{\partial I_{1}}{\partial z}} + {K_{2}\frac{\partial I_{2}}{\partial z}} + {K_{3}\frac{\partial I_{3}}{\partial z}}}{{K_{1}I_{1}} + {K_{2}I_{2}} + {K_{3}I_{3}}}}} & (16)\end{matrix}$

[0080] Where $\begin{matrix}{K_{1} \equiv {{\frac{\partial I_{3}}{\partial x}\frac{\partial I_{2}}{\partial y}} - {\frac{\partial I_{2}}{\partial x}\frac{\partial I_{3}}{\partial y}}}} \\{K_{2} \equiv {{\frac{\partial I_{1}}{\partial x}\frac{\partial I_{3}}{\partial y}} - {\frac{\partial I_{3}}{\partial x}\frac{\partial I_{1}}{\partial y}}}} \\{K_{3} \equiv {{\frac{\partial I_{2}}{\partial x}\frac{\partial I_{1}}{\partial y}} - {\frac{\partial I_{1}}{\partial x}\frac{\partial I_{2}}{\partial y}}}}\end{matrix}$

[0081] Next, a phase information restoring method according to a secondembodiment of the present invention will be described, referring toFIGS. 1, 3, and 4. FIG. 4 is an explanatory diagram of the phaseinformation restoring method according to the embodiment of the presentinvention and shows a condition in which X-ray imaging is performed inthe imaging unit. In the phase information restoring method according tothe embodiment, a visible image is constructed on the basis of imageinformation representing four diffraction fringe images taken with animaging distance changed.

[0082] First, at step S1, X-ray imaging is performed. The sensor 12 ispositioned at the position where the imaging distance is z₁ and theobject 10 is irradiated with an X-ray as shown in FIG. 4 so as toperform the X-ray imaging. Then, the sensor 12 moved to the positionwhere the imaging distance is z₂ and the X-ray imaging is similarlyperformed. Further, the X-ray imaging is repeated with the sensorpositioned at the imaging distances of z₃ and z₄. Thereby, the imageinformation representing diffraction fringe images are obtained.

[0083] By the X-ray imaging at step S1, detection data I₁(x,y), I₂(x,y),I₃(x,y), and I₄(x,y) are sequentially input to the phase informationrestoring apparatus 2 and stored in the storage unit 21. Here, thedetection data I₁(x,y) represents intensity of the diffracted light atthe position (x,y) on a plane at the imaging distance of z₁. Thedetection data I₂(x,y) to I₄(x,y) similarly represent intensity asabove.

[0084] Next, at step S2, the differential processing unit 22 obtains adifferential coefficient between detection data I_(N) and detection dataI_(N+1) using the following expression (17), where N=1, 2, and 3.$\begin{matrix}{\frac{\partial{I_{N}\left( {x,\quad y} \right)}}{\partial z} = \frac{{I_{N + 1}\left( {x,\quad y} \right)} - {I_{N}\left( {x,\quad y} \right)}}{z_{N + 1} - z_{N}}} & (17)\end{matrix}$

[0085] The processing at steps S3-S8 are the same as that described inthe first embodiment of the present invention.

[0086] Although, in the embodiment, phase restoration is performed byusing three differential coefficients obtained from four interferencefringe images taken while changing the imaging distance, the phaserestoration may be performed on the basis of four or more differentialcoefficients by using five or more interference fringe images.

[0087] Next, a phase information restoring apparatus according to athird embodiment of the present invention will be described. FIG. 5 is ablock diagram showing an X-ray imaging system including thephase:information restoring apparatus according to the embodiment of thepresent invention. This X-ray imaging system has a phase informationrestoring apparatus 7 for generating image data on the basis ofdetection data output from the imaging unit 1. Other construction is thesame as that of the X-ray imaging system shown in FIG. 1.

[0088] The phase information restoring apparatus 7 has a first storageunit 31 for temporarily storing the detection data output from theimaging unit 1, a differential processing unit 32 for obtaining adifferential coefficient between detection data at different imagingdistances, a Laplacian processing unit 33 for calculating a valuecorresponding to a Laplacian of phase, an inverse Laplacian processingunit 34 for performing inverse Laplacian operation for phaserestoration, a back propagation processing unit 35 for obtaining phaseinformation at a position of an object on the basis of the restoredphase information, detection data, and an imaging distance, a secondstorage unit 36 for temporarily storing the phase information at theposition of the object obtained in the back propagation processing unit35, an average processing unit 37 for averaging plural pieces of phaseinformation at the position of the object, an image processing unit 38for generating image data based on the averaged phase information, and acontrol unit 39 for controlling the above respective units 31-38 and theimaging distance in the imaging unit 1. The phase information restoringapparatus 7 may be configured with a digital circuit or software and aCPU. In the latter case, the control unit 39 including the CPU processesthe detection data on the basis of a phase information restorationprogram recorded on a recording medium 40. As the recording medium 40, aflexible disk, a hard disk, an MO, an MT, a RAM, a CD-ROM, a DVD-ROM,etc. are applicable.

[0089] Next, a principle of a phase information restoring methodaccording to the present invention will be described. The phaseinformation restoring method according to the present invention is amethod of constructing a visible image by the phase-contrast method, andthe phase restoration is performed on the basis of plural diffractionfringe images obtained with respect to an object by using the basicexpression of phase restoration, TIE (transport of intensity equation).

[0090] TIE expressed by the following expression (18) is transformed soas to obtain expression (19). $\begin{matrix}{{{- \kappa}\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z}} = {\nabla{\cdot \left\{ {{I\left( {x,\quad y} \right)}{\nabla{\varphi \left( {x,\quad y} \right)}}} \right\}}}} & (18) \\{{{- \kappa}\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z}} = {{{I\left( {x,\quad y} \right)}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}}} + {{\nabla{I\left( {x,\quad y} \right)}} \cdot {\nabla{\varphi \left( {x,\quad y} \right)}}}}} & (19)\end{matrix}$

[0091] Where I(x,y) is detection data representing intensity ofdiffracted light at a position (x,y) on a plane at a distance of z fromthe object.

[0092] In expression (19), approximating the second term ∇I(x,y)·∇φ(x,y)included in the right side to zero, the TIE approximation expression(20) is obtained. $\begin{matrix}{\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z} \cong {{- \frac{I\left( {x,\quad y} \right)}{\kappa}}{\nabla^{2}{\varphi \left( {x,\quad y} \right)}}}} & (20)\end{matrix}$

[0093] The phase information restoring apparatus according to theembodiment is for obtaining the phase used for generating image data byrestoring plural phases using the above TIE approximation expression(20) and averaging the restored phases.

[0094] Next, referring to FIGS. 2, 5, and 6, the phase informationrestoring method according to the third embodiment of the presentinvention will be described. FIG. 6 is a flowchart showing the phaseinformation restoring method according to the third embodiment of thepresent invention. In the embodiment, a visible image is constructed byusing detection data representing six diffraction fringe images takenwhile changing the imaging distance as shown in FIG. 2.

[0095] First, at step S10, X-ray imaging is performed. The sensor 12 ispositioned at the position where the imaging distance is z₁ and theobject 10 is irradiated with an X-ray so as to perform the X-rayimaging. Then, the sensor 12 moved to the position where the imagingdistance is (z₁+Δz₁) and the X-ray imaging is performed. Similarly, theX-ray imaging is repeated with the sensor 12 positioned at the imagingdistances of z₂, (z₂+Δz₂), z₃, and (z₃+Δz₃). Thereby, the detection datarepresenting diffraction fringe images are obtained.

[0096] By the X-ray imaging at step S10, the detection data I₁(x,y),I₁′(x,y), I₂(x,y), I₂′(x,y), I₃(x,y), and I₃′(x,y) are sequentiallyinput to the phase information restoring apparatus 7. Here, thedetection data I₁(x,y) represents intensity of the diffracted light atthe position (x,y) on a plane at the imaging distance of z₁. Similarly,the detection data I₁′(x,y), I₂(x,y), I₂′(x,y), I₃(x,y), and I₃′(x,y)represent intensity of the diffracted light at the positions (x,y) onplanes at the imaging distances of (z₁+Δz₁), z₂, (z₂+Δz₂), z₃, and(z₃+Δz₃), respectively. The detection data are sequentially stored inthe first storage unit 31 of the phase information restoring apparatus7.

[0097] Next, at steps S11-S13, the phase information restoring apparatus7 restores a phase at the position of the sensor on the basis of thedetection data stored in the first storage unit 31.

[0098] First, at step S11, the differential processing unit 32 obtains adifferential coefficient between detection data I_(N) and detection dataI_(N)′ using the following expression (21), where Δz_(N)=z_(N)′−z_(N)and N=1, 2, and 3. $\begin{matrix}{\frac{\partial{I_{N}\left( {x,\quad y} \right)}}{\partial z} = \frac{{I_{N}^{\prime}\left( {x,\quad y} \right)} - {I_{N}\left( {x,\quad y} \right)}}{\Delta \quad z_{N}}} & (21)\end{matrix}$

[0099] Then, at step S12, the Laplacian processing unit 33 obtainsLaplacian f(x,y)=∇²φ(x,y) of a phase on the basis of the differentialcoefficient obtained at step S11 and the detection data stored in thefirst storage unit 31, using the following expression (22).$\begin{matrix}{{f\left( {x,\quad y} \right)} = {{- \frac{\kappa}{I_{N}\left( {x,\quad y} \right)}}\frac{\partial{I\left( {x,\quad y} \right)}}{\partial z}}} & (22)\end{matrix}$

[0100] Here, in expression (22), although the differential coefficientis divided by the detection data I_(N)(x,y) at a shorter imagingdistance, it may be divided by the detection data I_(N)′(x,y) at alonger imaging distance or by different detection data from that usedwhen obtaining the differential coefficient. Alternatively, thedifferential coefficient may be divided by detection data performed withLPF (low pass filter) processing.

[0101] Further, at step S13, the inverse Laplacian processing unit 34performs inverse Laplacian operation on the Laplacian f(x,y)=∇²φ(x,y) ofthe phase obtained at step S12 so as to obtain phase φ(x,y). Note thatthe inverse Laplacian operation in the inverse Laplacian processing unit34 is the same as that described using FIG. 3 in the first embodiment ofthe present invention.

[0102] Next, at steps S14-S16, the back propagation processing unit 35restores a phase of the X-ray just after transmitted through the objecton the basis of the restored phase, the detection data I₁, I₂, and I₃stored in the storage unit 31, and the imaging distances z₁, z₂, and z₃.Hereinafter, a phase etc. of an X-ray just after transmitted through anobject is referred to as a phase etc. at the position of the object inrelation to a phase etc. of the X-ray at the imaging distance of z_(N).

[0103] First, at step S14, the back propagation processing unit 35obtains X-ray wave ψ_(N)(x,y) at the imaging distance of z_(N) on thebasis of the phase Φ_(N)(x,y) restored at step S13 and the detectiondata I_(N)(x,y) stored in the first storage unit 31, using the followingexpression (23).

ψ_(N)(x,y)={square root}{square root over (I _(N)(x,y))} exp[iφ_(N)(x,y)   (23)

[0104] Where “i” denotes imaginary unit.

[0105] Next, at step S15, the back propagation processing unit 35obtains X-ray wave ψ_(N→0)(x,y) at the position of the object on thebasis of the X-ray wave ψ_(N)(x,y) obtained at step S14 using thefollowing expression (24).

ψ_(N→0)(x,y)=h _(−ZN)(x,y)*ψ_(N)(x,y)   (24)

[0106] Where “*” denotes convolution and${h_{z}\left( {x,\quad y} \right)} = {\frac{1}{i\quad \lambda \quad z}^{\frac{i\quad \pi}{\lambda \quad z}{({x^{2} + y^{2}})}}}$

[0107] Further, at step S16, the back propagation processing unit 35calculates phase φ_(N→0)(x,y) at the position of the object on the basisof the X-ray wave ψ_(N→0)(x,y) at the position of the object obtained atstep S15, using the following expression (25). The calculated phaseφ_(N→0)(x,y) is sequentially stored in the second storage unit 36.$\begin{matrix}{{\varphi_{N\rightarrow 0}\left( {x,\quad y} \right)} = {\tan^{- 1}\left\lbrack \frac{{Im}\left\lbrack {\Psi_{N->0}\left( {x,\quad y} \right)} \right\rbrack}{{Re}\left\lbrack {\Psi_{N->0}\left( {x,\quad y} \right)} \right\rbrack} \right\rbrack}} & (25)\end{matrix}$

[0108] Where Re [ ] and Im [ ] are functions for obtaining the real partand the imaginary part, respectively.

[0109] Next, at step S17, the average processing unit 37 calculatesaverage phase φ_(O)(x,y) at the position of the object on the basis ofthe phase φ_(O)(x,y) at the position of the object stored in the secondstorage unit 36 using the following expression (26). $\begin{matrix}{{\varphi_{0}\left( {x,\quad y} \right)} = {\frac{1}{3}{\sum\limits_{\quad {N = {1,\quad 2,\quad 3}}}^{\quad}\quad {\varphi_{N->0}\left( {x,\quad y} \right)}}}} & (26)\end{matrix}$

[0110] Then, at step S18, the image processing unit 38 generates imagedata on the basis of the average phase φ_(O)(x,y). That is, the imageprocessing unit 38 converts the average phase φ_(O)(x,y) in each pixelinto data representing brightness and performs necessary imageprocessing such as gradation processing and interpolation processing.

[0111] At step S19, the display unit 3 and the output unit 4 displays avisible image on a screen or a film on the basis of the image datagenerated as described above.

[0112] Although, in the embodiment, the method of restoring phase byusing three differential coefficients obtained from six interferencefringe images taken while changing the imaging distance is described,the phase restoration may be performed on the basis of two differentialcoefficients, or the images used when obtaining different differentialcoefficients may be duplicated.

[0113] Next, a phase information restoring apparatus according to afourth embodiment of the present invention will be described. FIG. 7 isa block diagram showing an X-ray imaging system including the phaseinformation restoring apparatus according to the fourth embodiment ofthe present invention. This X-ray imaging system includes a phaseinformation restoring apparatus 8 instead of the phase informationrestoring apparatus 7 in FIG. 5. Other construction is the same as thatin FIG. 5.

[0114] The phase information restoring apparatus 8 has a first storageunit 31 for temporarily storing the detection data output from theimaging unit 1, a differential processing unit 32 for obtaining adifferential coefficient between detection data at different imagingdistances, a Laplacian processing unit 33 for calculating a valuecorresponding to a Laplacian of phase, an inverse Laplacian processingunit 34 for performing an inverse Laplacian operation for performingphase restoration, a second storage unit 36 for temporarily storing thephase information at the position of the sensor output from the inverseLaplacian processing unit 34, an average processing unit 37 foraveraging plural pieces of phase information, an image processing unit38 for generating image data based on the averaged phase information,and a control unit 39 for controlling the above respective units 31-38and the imaging distance in the imaging unit 1. The phase informationrestoring apparatus 8 may be configured with a digital circuit orsoftware and a CPU.

[0115] Next, a phase information restoring method according to thefourth embodiment of the present invention will be described, referringto FIGS. 2, 7, and 8. FIG. 8 is a flowchart showing the phaseinformation restoring method according to the fourth embodiment of thepresent invention. In the embodiment, a visible image is constructed byusing image information representing six diffraction fringe images takenwhile changing the imaging distance. The method is characterized byusing TIE approximation expression expressed by expression (20) andperforming appropriate approximation when constructing an operationexpression so as to perform operation easily and speedy.

[0116] First, at step S20, X-ray imaging is performed. The sensor 12 ispositioned at the position where the imaging distance is z₁ and theobject 10 is irradiated with an X-ray as shown in FIG. 2 so as toperform the X-ray imaging. Subsequently, the sensor 12 moved to theposition where the imaging distance is (z₁+Δz₁) and the X-ray imaging issimilarly performed. Further, the X-ray imaging is repeated with thesensor 12 positioned at the imaging distances of z₂, (z₂+Δz₂), z₃, and(z₃+Δz₃). Thereby, the image information representing diffraction fringeimages are obtained.

[0117] By the X-ray imaging at step S20, the detection data I₁(x,y),I₁′(x,y), I₂(x,y), I₂′(x,y), I₃(x,y), and I₃′(x,y) are sequentiallyinput to the phase information restoring apparatus 8. Here, thedetection data I₁(x,y) represents intensity of the diffracted light atthe position (x,y) on a plane at the imaging distance of z₁. Similarly,the detection data I₁′(x,y), I₂(x,y), I₂′(x,y), I₃(x,y), and I₃′(x,y)represent intensity of the diffracted light at the positions (x,y) onplanes at the imaging distances of (z₁+Δz₁), z₂, (z₂+Δz₂), z₃, and(z₃+Δz₃), respectively. The detection data are sequentially stored inthe first storage unit 31 of the phase information restoring apparatus8.

[0118] Next, at steps S21-S23, the phase information restoring apparatus8 restores a phase at the position of the sensor on the basis of thedetection data stored in the first storage unit 31.

[0119] First, at step S21, the differential processing unit 32 obtains adifferential coefficient between detection data I_(N) and detection dataI_(N)′.

[0120] Then, at step S22, the Laplacian processing unit 33 obtainsLaplacian f(x,y)=∇²φ(x,y) of a phase on the basis of the differentialcoefficient obtained at step S21 and the detection data stored in thefirst storage unit 31.

[0121] Further, at step S23, the inverse Laplacian processing unit 34performs an inverse Laplacian operation on the Laplacian f(x,y)=∇²φ(x,y)of the phase obtained at step S22 so as to calculate phase φ(x,y). Thecalculated phase φ(x,y) is sequentially stored in the second storageunit 36.

[0122] Next, at step S24, the average processing unit 37 calculatesaverage phase φ_(O)(x,y) based on the phase φ_(N)(x,y) at the positionof the sensor stored in the second storage unit 36.

[0123] Further, at step S25, the image processing unit 38 generatesimage data based on the average phase φ_(O)(x,y). That is, the imageprocessing unit 38 converts the average phase φ_(O)(x,y) in each pixelinto data representing brightness and performs necessary imageprocessing such as gradation processing and interpolation processing.

[0124] At step S26, the display unit 3 and the output unit 4 display avisible image on a screen or a film on the basis of the image datagenerated as described above.

[0125] In the embodiment, the phases φ₁ to φ₃ at different imagingdistances are averaged as at step S24. In the strict sense, these phasesφ₁ to φ₃ include differences in accordance with the changes in theimaging distances in relation to the phase φ_(O) at the position of theobject. However, when a light source such as a synchrotron radiationsource that generates a highly directional beam is used, these phasesφ₁, φ₂, and φ₃ can be approximated equal to the phase φ_(O) at theposition of the object. Further, averaging the phases φ₁ to φ₃ cancancel errors and bring the averaged phase closer to the real phaseφ_(O).

[0126] Although, in the first to fourth embodiments of the presentinvention described above, X-rays are used when imaging is performed onan object, any beam other than X-rays that can be transmitted throughthe object and form diffraction images, such as particle beams includingan electron beam, may be used.

[0127] Further, in the first to fourth embodiments of the presentinvention, although a synchrotron radiation source is used when imagingis performed on an object, a light source generating beams other thansynchrotron radiation may be used. For example, an electron storage typehigh brightness hard X-ray generator, which has been developed byRitsumeikan University, can generate X-rays having as high brightnessand directivity as synchrotron radiation despite of its tabletop size.X-rays generated by this generator have coherency, and even though theX-rays have plural wavelengths, they can be monochromatized by combiningwith monochromatizing crystal. Furthermore, the radiation sourcedeveloped by The Femtosecond Technology Research Association (FESTA)generates ultrashort pulse high-brightness X-rays based on a principleof backward Compton scattering. This ray source is compact and portable,and can generate X-rays having not only coherency but also highdirectivity and monochromaticity. Note that, if a point light source isused as a light source, it is desirable to correct the detection data toinclude magnification when performing data processing in the phaseinformation restoring apparatus.

[0128] Next, a modified example of the X-ray imaging system includingthe phase information restoring apparatus according to the first tofourth embodiments of the present invention will be described. The X-rayimaging system shown in FIG. 9 has a reading unit 5 and an imaging unit6 instead of the imaging unit 1 in the X-ray imaging system shown inFIG. 1. Other construction is the same as that of the X-ray imagingsystem shown in FIG. 1.

[0129] In the imaging unit 6, as a screen used for recording imageinformation, a photostimulable phosphor sheet (recording sheet) is usedinstead of the sensor 12 in the imaging unit 1 shown in FIG. 2.

[0130] The photostimulable phosphor (storage phosphor) is a materialthat, when applied with radiation, a part of the radiation energy isstored therein, and when applied with excitation light such as visiblelight afterward, light is photostimulably emitted in accordance with thestored energy. When a radiation image of an object such as a human bodyis taken and recorded on the sheet applied with the photostimulablephosphor, and scanned by the excitation light such as laser light,stimulated fluorescent light is generated. Therefore, detection data canbe obtained by reading the light photoelectrically. After the detectiondata is appropriately processed, the radiation image can be displayed asa visible image by outputting to a display such as a CRT or printing outon a film by a laser printer etc.

[0131] The reading unit 5 shown in FIG. 9 is used for reading theradiation image recorded on the recording sheet. Here, referring to FIG.10, construction and operation of the reading unit 5 will be described.The recording sheet 50 on which image information has been recorded isset in a predetermined position in the reading unit 5. The recordingsheet 50 is carried in Y-direction by a sheet carrying means 52 drivenby a motor 51. On the other hand, a beam L1 oscillating from the lasersource 53 is reflected and deflected by a rotating polygon mirror 55driven by a motor 54 and rotating at high speed in a direction of anarrow, and passes through a convergent lens 56. Then, the beam L1changes its optical path by the mirror 57 and scans the recording sheet50 in X-direction. By the scanning, excitation light L2 is applied tothe recording sheet 50, and stimulated fluorescent light L3 havingintensity in accordance with the stored and recorded radiation imageinformation is emitted from the applied part. The stimulated fluorescentlight L3 is guided by the optical guide 58 and photoelectricallydetected by a photomultiplier 59. An analogue signal output from thephotomultiplier 59 is amplified by an amplifier 60 and digitized by anA/D converter 61. The detection data output from the A/D converter 61 isinput to the phase information restoring apparatus 2.

[0132] Image information representing plural interference fringe imagesobtained at different imaging distances can be obtained by performingradiation imaging with the imaging distance changed and using pluralrecording sheets in the imaging unit 6, and reading image informationfrom the respective recording sheets in the reading unit 5. The phaseinformation restoring apparatus 2 performs phase restoration based onthe image information and generates image data. The processing in thephase information restoring apparatus 2 is the same as that describedusing FIG. 3.

[0133] The X-ray imaging system shown in FIGS. 5 and 7 can also bemodified into an X-ray imaging system using a photostimulable phosphorsheet similarly to that shown in FIG. 9.

[0134] As described above, according to the present invention, ahigh-accuracy phase restoration can be easily performed by minimizingapproximation in TIE and performing operation using matrices. Thus, avisible image of good quality can be obtained by the phase-contrastmethod.

[0135] Further, according to the present invention, phase information ofhigh accuracy can be obtained by averaging the plural restored phases toobtain the phase used as image data. Therefore, a visible image of goodquality in which noise is cancelled can be obtained by using the abovephase information.

1. A phase information restoring method of restoring phase informationof radiation transmitted through an object on the basis of an imagesignal obtained by detecting intensity of the radiation transmittedthrough the object, said method comprising the steps of: (a) obtainingat least three first differential signals representing differentialsbetween one image signal and another image signal on the basis of atleast four image signals obtained by detecting intensity of radiation onat least four planes that are parallel and positioned at differentdistances from the object, said at least four image signals respectivelyrepresenting radiation image information on said at least four planes;(b) obtaining second differential signals and third differential signalsrepresenting differentials between image signals relative to twodirections orthogonal to each other within said planes with respect toat least three image signals; (c) obtaining a Laplacian of phase on thebasis of the at least three image signals and at least three sets of thefirst to third differential signals; and (d) performing inverseLaplacian operation on the Laplacian of phase so as to restore phaseinformation.
 2. A phase information restoring method according to claim1, wherein step (c) includes obtaining the Laplacian of phase by usingan inverse matrix of a matrix having elements of the at least threeimage signals and at least three sets of the second and thirddifferential signals.
 3. A phase information restoring method ofrestoring phase information of radiation transmitted through an objecton the basis of an image signal obtained by detecting intensity of theradiation transmitted through the object, said method comprising thesteps of: (a) obtaining plural differential signals representingdifferentials between one image signal and another image signal on thebasis of at least three image signals obtained by detecting intensity ofradiation on at least three planes that are positioned at differentdistances from the object, said at least three image signalsrespectively representing radiation image information on said at leastthree planes; (b) respectively obtaining Laplacian of phases on thebasis of said plural differential signals and one of said at least threeimage signals; (c) performing inverse Laplacian operation on theLaplacian of phases so as to obtain plural phases respectively; (d)calculating an average value of the plural phases obtained at step (c).4. A phase information restoring method according to claim 3, whereinstep (c) includes respectively calculating waves of the radiation on theplanes on the basis of the image signals used when obtaining therespective differential signals at step (a) and the plural phasesrestored by using the respective differential signals, converting thewaves of the radiation on said planes into plural waves on apredetermined plane by back propagation calculation, and obtainingplural phases on said predetermined plane.
 5. A phase informationrestoring apparatus for restoring phase information of radiationtransmitted through an object on the basis of an image signal obtainedby detecting intensity of the radiation transmitted through the object,said apparatus comprising: differential processing means for obtainingat least three first differential signals representing differentialsbetween one image signal and another image signal on the basis of atleast four image signals obtained by detecting intensity of radiation onat least four planes that are parallel and positioned at differentdistances from the object, said at least four image signals respectivelyrepresenting radiation image information on said at least four planes,and obtaining second differential signals and third differential signalsrepresenting differentials between image signals relative to twodirections orthogonal to each other within said planes with respect toat least three image signals; Laplacian processing means for obtaining aLaplacian of phase on the basis of the at least three image signals andat least three sets of the first to third differential signals; inverseLaplacian processing means for performing inverse Laplacian operation onthe Laplacian of phase so as to restore phase information.
 6. A phaseinformation restoring apparatus according to claim 5, wherein saidLaplacian processing means obtains the Laplacian of phase by using aninverse matrix of a matrix having elements of the at least three imagesignals and at least three sets of the second and third differentialsignals.
 7. A phase information restoring apparatus for restoring phaseinformation of radiation transmitted through an object on the basis ofan image signal obtained by detecting intensity of the radiationtransmitted through the object, said apparatus comprising: differentialprocessing means for obtaining plural differential signals representingdifferentials between one image signal and another image signal on thebasis of at least three image signals obtained by detecting intensity ofradiation on at least three planes that are positioned at differentdistances from the object, said at least three image signalsrespectively representing radiation image information on said at leastthree planes; Laplacian processing means for respectively obtainingLaplacian of phases based on said plural differential signals and one ofsaid at least three image signals; inverse Laplacian processing meansfor performing inverse Laplacian operation on the Laplacian of phases soas to obtain plural phases respectively; average processing means forcalculating an average value of the plural phases obtained in saidinverse Laplacian processing means.
 8. A phase information restoringapparatus according to claim 7, wherein said inverse Laplacianprocessing means respectively calculates waves of the radiation on theplanes on the basis of the image signals used when obtaining therespective differential signals in said differential processing meansand the plural phases restored by using the respective differentialsignals, converts the waves of the radiation on said planes into pluralwaves on a predetermined plane by back propagation calculation, andobtains plural phases on said predetermined plane.
 9. A phaseinformation restoring program to be used for restoring phase informationof radiation transmitted through an object on the basis of an imagesignal obtained by detecting intensity of the radiation transmittedthrough the object, said program allowing a CPU to execute theprocedures of: obtaining at least three first differential signalsrepresenting differentials between one image signal and another imagesignal on the basis of at least four image signals obtained by detectingintensity of radiation on at least four planes that are parallel andpositioned at different distances from the object, said at least fourimage signals respectively representing radiation image information onsaid at least four planes; obtaining second differential signals andthird differential signals representing differentials between imagesignals relative to two directions orthogonal to each other within saidplanes with respect to at least three image signals; obtaining aLaplacian of phase on the basis of the at least three image signals andat least three sets of the first to third differential signals; andperforming inverse Laplacian operation on the Laplacian of phase so asto restore phase information.
 10. A phase information restoring programto be used for restoring phase information of radiation transmittedthrough an object on the basis of an image signal obtained by detectingintensity of the radiation transmitted through the object, said programallowing a CPU to execute the procedures of: (a) obtaining pluraldifferential signals representing differentials between one image signaland another image signal on the basis of at least three image signalsobtained by detecting intensity of radiation on at least three planesthat are positioned at different distances from the object, said atleast three image signals respectively representing radiation imageinformation on said at least three planes; (b) respectively obtainingLaplacian of phases on the basis of said plural differential signals andone of said at least three image signals; (c) performing inverseLaplacian operation on the Laplacian of phases so as to obtain pluralphases respectively; (d) calculating an average value of the pluralphases obtained in procedure (c).